Very, very reminiscent of the Hendershot device. A couple of suggestions:
1. Doing nothing else, turn the device 90 degrees and observe any difference to determine if variation is related to earth's magnetic field.
2. Do the same thing with and without the the nearby pulse motor and with the pulse motor rotated or not.
3. Do all of above at 180 degree rotation, i.e. north south poles swapped.
4. Look at the Wesley Gary Magnetic Motor stuff to see if anything looks related, i.e. the "neutral zone". Along those lines, put spacers of varying thicknesses between the horseshoe Alnico and the coil device to see if you can find a sweet spot. It may be that the sound is wiggling that within that "neutral zone".
Someone mentioned in an earlier post to look at the piezo's as capacitors. That's what tripped the switch in my head about the Hendershot device. The finicky nature of this device is also similar to the Hendershot stories.